A classical chirp sinusoid is represented simply byƒ(t)=sin(πrt2)  Eq. (1)where the time origin is chosen to be the point where the chirp waveform passes through zero frequency. The derivative of the phase argument is 2πrt, indicating that the constant r expresses a phase acceleration, or frequency ramp rate in frequency per unit time. If r is large, then the function ƒ(t) appears very non-sinusoidal, moving in instantaneous frequency from DC to high frequency in a short time. However, if r is small, then the function ƒ(t) appears to approximate a sinusoid of constant frequency for relatively-long time periods.
The function ƒ(t) may equally well be written asƒ(t)=sin(πrt2 mod 2π)  Eq. (2)where mod represents a modulus operation. If the angular units of the argument are changed to those where the quantity
  2  rrepresents a full period, and furthermore the time unit is taken to be that of the sampling frequency ƒs, then the function can be written as the sampled functionƒt=sinm(t2 mod m)  Eq. (3)where in general m=2ƒs2/r, representing the angle which corresponds to 2π radians. The integer t represents the tth sampling instant, and the modified function sinm( ) indicates that the units of the argument are such that the modulus m represents 2π radians.
It is assumed that r and ƒs are such that m is an integer. For a fixed ƒs, this assumption amounts to a quantization constraint in the frequency ramp rate r. For typical applications, choices for r remain practically continuous even with fixed ƒs. The choices are fully continuous if ƒs need not be exactly fixed at design time. For a frequency ramp rate r in the range of 1 MHz per microsecond, played at a typical sample rate of 1 Gs/s, m=2,000,000. The next available integral choice for m (2,000,001) would give a ramp rate slightly smaller than 1 MHz per microsecond (smaller by 1 part in 2 million).
The phase of the sinusoid function, a=t2 mod m, in equation 3, generates a sequence of integers. Each of these integers is called a quadratic residue of m. Some older textbooks such as Charles Varden Eynden, Number Theory: An Introduction to Proof, International Textbook Company, 1970, further stipulate that a quadratic residue must also be coprime with the modulus m. The terms “coprime” and “relatively prime” both describe a set of numbers that share no common factors. However, the broader, more-modern definition is used herein. Number theory texts have many pages devoted to the properties of this deceptively simple function. It is, for example, relatively easy to find a given an integer t. However, it is nontrivial, in the general case, to find t when a is given (or even to determine whether there exists such a t).
For example, if m=16, then the generated sequence of quadratic residues begins with 0,1,4,9,0,9,4,1,0,1,4,9,0,9,4,1, . . . . The sequence appears to repeat indefinitely. Only 4 quadratic residues appear to exist modulo 16, when one might expect to observe as many as 16.
Conventional high speed chirp waveform signal generators are complex and have relatively high power requirements. There is a need for improved chirp waveform signal generators. In particular, there is a need for chirp waveform signal generators that operate at high speed with relatively low power requirements.